An essay on some problems of approximation theory

arXiv:math/0301380v1[math.CA]31Jan2003Anessayonsomeproblemsofapproximationtheory∗†A.G.RammMathematicsDepartment,KansasStateUniversity,Manhattan,KS66506-2602,[e-mail]onsofapproximationtheoryarediscussed:1)canoneapproxi-matestablyinL∞normf′givenapproximationfδ,kfδ−fkL∞δ,ofanunknownsmoothfunctionf(x),suchthatkf′(x)kL∞≤m1?2)canoneapproximateanarbitraryf∈L2(D),D⊂Rn,n≥3,isaboundeddomain,bylinearcombinationsoftheproductsu1u2,whereum∈N(Lm),m=1,2,LmisaformallinearpartialdifferentialoperatorandN(Lm)isthenull-spaceofLminD,N(Lm):={w:Lmw=0inD}?3)canoneapproximateanarbitraryL2(D)functionbyanentirefunctionofexponentialtypewhoseFouriertransformhassupportinanarbitrarysmallopenset?Isthereananalyticformulaforsuchanapproximation?1IntroductionInthisessayIdescribeseveralproblemsofapproximationtheorywhichIhavestudiedandwhichareofinterestbothbecauseoftheirmathematicalsignificanceandbecauseoftheirimportanceinapplications.1.1ThefirstquestionIhaveposedaround1966.Thequestionis:supposethatf(x)isasmoothfunction,sayf∈C∞(R),whichisT-periodic(justtoavoidadiscussionofitsbehaviorneartheboundaryofaninterval),andwhichisnotknown;assumethatitsδ-approximationfδ∈L∞(R)isknown,kfδ−fk∞δ,wherek·k∞istheL∞(R)norm.Assumealsothatkf′k∞≤m1∞.CanoneapproximatestablyinL∞(R)thederivativef′,giventheabovedata{δ,fδ,m1}?∗keywords:stabledifferentiation,approximationtheory,propertyC,ellipticequations,Runge-typetheorems,scatteringsolutions†Mathsubjectclassification:41-XX,30D20,35R25,35J10,35J05,65M30Byapossibilityofastableapproximation(estimation)ImeantheexistenceofanoperatorLδ,linearornonlinear,suchthatsupf∈C∞(R)kf−fδk∞≤δ,kf′k∞≤m1kLδfδ−f′k∞≤η(δ)→0asδ→0,(1.1)whereη(δ)0issomecontinuousfunction,η(0)=0.Withoutlossofgeneralityonemayassumethatη(δ)ismonotonicallygrowing.In1962-1966therewasgrowinginteresttoill-posedproblems.Variationalregulariza-tionwasintroducedbyD.L.Phillips[2]in1962andayearlaterbyA.N.Tikhonov[30]in1963.Itwasappliedin[1]in1966totheproblemofstablenumericaldifferentiation.Themethodforstabledifferentiationproposedin[1]wascomplicated.Ithenproposedandpublishedin1968[3]theideatouseadivideddifferenceforstabledifferentiationandtousethestepsizeh=h(δ)asaregularizationparameter.Ifkf′′k≤m2,thenh(δ)=q2δm2,andifonedefines([3]):Lδfδ:=fδ(x+h(δ))−fδ(x−h(δ))2h(δ),h(δ)=r2δm2,(1.2)thenkLδfδ−f′(x)k∞≤p2m2δ:=ε(δ).(1.3)ItturnsoutthatthechoiceofLδ,madein[3],thatis,Lδdefinedin(1.2),isthebestpossibleamongalllinearandnonlinearoperatorsTwhichapproximatef′(x)giventheinformation{δ,m2,fδ}.Namely,ifK(δ,mj):={f:f∈Cj(R),mj∞,kf−fδk∞≤δ},andmj=kf(j)k∞,theninfTsupf∈K(δ,m2)kTfδ−f′k≥p2m2δ.(1.4)Onecanfindaproofofthisresultandmoregeneralonesin[4],[5],[8],[9]andvariousapplicationsoftheseresultsin[4]-[7].Theideaofusingthestepsizehasaregularizationparameterbecamequitepopularafterthepublicationof[3]andwasusedbymanyauthorslater.In[27]formulasaregivenforasimultaneousapproximationoffandf′.1.2Thesecondquestion,thatIwilldiscuss,isthefollowingone:canoneapproximate,withanarbitraryaccuracy,anarbitraryfunctionf(x)∈L2(D),orinLp(D)withp≥1,byalinearcombinationoftheproductsu1u2,whereum∈N(Lm),m=1,2,Lmisaformallinearpartialdifferentialoperator,andN(Lm)isthenull-spaceofLminD,N(Lm):={w:Lmw=0inD}?2ThisquestionhasledmetothenotionofpropertyCforapairoflinearformalpartialdifferentialoperators{L1,L2}.Letusintroducesomenotations.LetD⊂Rn,n≥3,beaboundeddomain,Lmu(x):=P|j|≤Jmajm(x)Dju(x),m=1,2,jisamultiindex,Jm≥1isaninteger,ajm(x)aresomefunctionswhosesmoothnesspropertieswedonotspecifyatthemoment,Dju=∂ju∂xj11...∂xjnn,|j|=j1+···+jn.DefineNm:=ND(Lm):={w:Lmw=0inD},wheretheequationisunderstoodinthesenseofdistributiontheory.Considerthesetofproducts{w1w2},wherewm∈Nmandweusealltheproductswhicharewell-defined.IfLmareellipticoperatorsandajm(x)∈Cγ(Rn),thenbyellipticregularitythefunctionswm∈Cγ+Jmandthereforetheproductsw1w2arewelldefined.Definition1.1.Apair{L1,L2}haspropertyCifandonlyiftheset{w1w2}∀wm∈NmistotalinLp(D)forsomep≥1.Inotherwords,iff∈Lp(D),thenZDf(x)w1w2dx=0,∀wm∈Nm⇒f=0,(1.5)where∀wm∈Nmmeansforallwmforwhichtheproductsw1w2arewelldefined.Definition1.2.Ifthepair{L,L}haspropertyCthenwesaythattheoperatorLhasthisproperty.FromthepointofviewofapproximationtheorypropertyCmeansthatanyfunctionf∈Lp(D)canbeapproximatedarbitrarilywellinLp(D)normbyalinearcombinationofthesetofproductsw1w2oftheelementsofthenull-spacesNm.Forexample,ifL=∇2thenN(∇2)isthesetofharmonicfunctions,andtheLaplacianhaspropertyCifthesetofproductsh1h2ofharmonicfunctionsistotal(complete)inLp(D).ThenotionofpropertyChasbeenintroducedin[10].Itwasdevelopedandwidelyusedin[10]-[21].Itprovedtobeaverypowerfultoolforastudyofinverseproblems[15]-[18],[20]-[21].UsingpropertyCtheauthorhasprovedin1987theuniquenesstheoremfor3Dinversescatteringproblemwithfixed-energydata[12],[13],[17],uniquenesstheoremsforinverseproblemsofgeophysics[12],[16],[18],andformanyotherinverseproblems[18].Theaboveproblemshavebeenopenforseveraldecades.1.3ThethirdquestionthatIwilldiscuss,dealswithapproximationbyentirefunctionsofexponentialtype.Thisquestionisquitesimplebuttheanswerwasnotcleartoengineers